Singular measures of piecewise smooth circle homeomorphisms with two break points

Let $T_{f}$ : S1 → S1 be a circle homeomorphism with two break points ab, cb that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã b, ĉb, which are the representative points ofab, cb in the interval $[0,1)$, and irrationalrotation number ρf. Suppose that $Df$ is absolutely continuous on every connected intervalof the set [0,1]\{ãb, ĉb}, that DlogDf ∈ L1([0,1]) and the product of the jump ratios of $ Df $ at thebreak points is nontrivial, i.e.$\frac{Df_{-}(\tilde{a}_{b})}{Df_{+}(\tilde{a}_{b})}\frac{Df_{-}(\tilde{c}_{b})}{Df_{+}(\tilde{c}_{b})}\ne1$.We prove, that the unique Tf - invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure onS1.

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